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10. The Method of Cluster Expansions
Cluster Expansion for a Classical Gas Virial Expansion of the Equation of State Evaluation of the Virial Coefficients General Remarks on Cluster Expansions Exact Treatment of the Second Virial Coefficient Cluster Expansion for a Quantum Mechanical System Correlations & Scattering
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Cluster expansions = Series expansion to handle inter-particle interactions
Applicability : Low density gases Poineers : Mayer : Classical statistics. Kahn-Uhlenbeck, Lee-Yang : Quantum statistics
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10.1. Cluster Expansion for a Classical Gas
Central forces : Partition function :
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where Configuration integral Non-interacting system ( uij = 0 ) : Let L-J potential
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Graphic Expansion All possible pairings 8-particle graphs : = =
factorized = =
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l - Cluster Each N-particle integral is represented by an N-particle graph. Graphs of the same topology but different labellings are counted as distinct. An l-cluster graph is a connected l-particle graph. ( Integral cannot be factorized. ) E.g., 5-cluster : = Integrals represented by l-clusters of the same topology has the same value. All possible 3-clusters : = =
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Cluster Integrals Cluster integral : Let = dimension of X.
X is dimensionless ru = range of u For a fixed r1 , is indep of V. is indep of size & shape of system
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Examples V(r1) = volume of gas using r1 as origin.
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ZN Let ml = # of l-cluster graphs for each N-particle graph
Let be the sum of all graphs that satisfy # of distinct ways to assign particles into is Let there be pl distinct ways to form an l-cluster, with each giving an integral Il j . Then the sum of all distinct products of ml of these l-clusters is The factor ml ! arises because the order of Il j within each product is immaterial.
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where
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Z, Z, F, P, n
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10.2. Virial Expansion of the Equation of State
Virial expansion for gases : Invert gives Mathematica
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In general : (see §10.4 for proof )
irreducible cluster integral ( dimensionless ) Irreducible means multiply-connected, i.e., more than one path connecting any two vertices. c.f.
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10.3. Evaluation of the Virial Coefficients
Lennard-Jones potential : minimum Precise form of repulsive part ( u > 0 ) not important. Can be replaced by impenetrable core ( u = r < r0 ). Precise form of attractive part ( u < 0 ) important : Useful adjustable form :
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a2 For : Bl are also called the virial coefficients
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van der Waals Equation for for c.f. van der Waals eq.
v0 = molecular volume see Prob 1.4 r0 = molecular diameter Condition ( dilute gas )
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B2 where Reduced Lennard-Jones potential
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Hard Sphere Gas Molecules = Hard spheres Step potential :
D = diameter of spheres D D Mathematica
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See Pathria, p.314 for values of a4 , a5 , a6 & P.
Mathematica See Pathria, p.314 for values of a4 , a5 , a6 & P. Approximate analytic form of the equation of state for fluids ( ) :
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10.4. General Remarks on Cluster Expansions
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Coefficients of Zjk in ( ... )l sum to 0.
Classical ideal gas : ( ... )l ~ sum of all possible l-clusters are independent of V ( ... )l V Rushbrooke :
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Semi-Invariants Constraint (l ) : Semi-Invariants Inversion :
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Proof of inversion QED
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A theorem due to Lagrange :
Solution x(z) to eq. is where
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constraint (j1) : Inversion due to Mayer : constraint (l1) :
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10.5. Exact Treatment of the Second Virial Coefficient
u(r) = 0 where Total Reduced Let
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Let spectrum of interacting system consist of a discrete (bounded states) part & a continuum (travelling states) part with DOS g().
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Unbounded states ( n > 0 )
where l = phase shift
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For the purpose of counting states ( to get g() ),
we discretize the spectrum by setting for some . For a given l , k l m is 2l+1 fold degenerate e/o means l in sum is even/odd for boson/fermion For u = 0 :
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Boson Fermion From § 7.1 & § 8.1 :
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b2(0) From § 5.5 : same as before
Alternatively, using the statistical potential from § 5.5
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Hard Sphere Gas In region where u = 0, Mathematica
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No bound states for hard sphere gas.
Mathematica
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